In this unit students are encouraged to abandon counting methods, in favour of using known doubles facts, to derive the answers to addition problems. Using doubles is more complex but more efficient.
In this unit students are encouraged to progress from using counting on or down methods, to the use of doubles and part-whole mental methods to speed up mental computation.
Examples of counting on or counting down methods are:
Examples of more efficient part-whole methods, which draw on doubling, are:
It is desirable for students to move from counting methods to part/whole methods as counting methods become too slow when working with larger and more complex numbers.
The progression from a counting method to the more abstract part/whole methods can be difficult for students. The idea of breaking numbers into suitable parts to help solve the problem is not initially obvious and may need to multiple opportunities for revisiting, modelling, and working through the strategy. For example students might model this problem on counters: Lucy has 7 orange sweets and buys 8 green sweets at the shop. How many has she got altogether?
A green sweet is removed to leave 7 in one pile and 7 in the other. Students, because they know their doubles, now say 7 + 7 = 14. Critically they must now add the removed sweet to 14 to give 15.
While this process often makes sense for students when materials are present they may experience difficulty when the materials are removed and they are asked to visualise the process. For example, to work out 9 cakes plus 8 cakes the students are asked to "see" without materials, that 9 contains 8 and 1, then to understand they use the known double 8 + 8 = 16, then they add the 1 to give 17.
The step to being able to work out addition problems by visualising is an important step along the way to being able to quickly and reliably use a "near doubles " strategy. Persistence and repeated practice is needed to help students make this transformation from needing materials to be able to image the answer.
The learning opportunities in this unit can be differentiated by providing or removing support to students and varying the task requirements. Ways to support students include:
To increase the relevance of this unit, the questions posed can be adapted to reflect the interests, experiences, and cultural backgrounds of your students. You could also encourage students ready for extension to create word problems that reflect a shared, relevant context and the mathematics introduced in the session. In turn, these problems could be used as practice tasks for the rest of the class.
Te reo Māori kupu such as uara tū (place value), rearua (double), and tatau (count) could be introduced in this unit and used throughout other mathematical learning.
You could also encourage students, who speak a language other than English at home, to share the words related to counting, adding, and doubling from their home language.
Pose addition problems that are near doubles, that is to say the two numbers to be added are almost equal. Encourage students to solve these problems with counters, first removing or adding counters to make the piles equal, then adding or removing counters to finish the problem.
Sarah has 6 sweets and Matua has 8 sweets.
Sarah complains it is not fair because Matua has got more sweets.
Matua gives a third student some of his sweets so Sarah and Matua are now equal
How many sweets does Matua give away? (2)
How many sweets do Sarah and Matua have now? (6+6=12)
How many sweets are there altogether (12+2=14)
Another strategy for solving the problem is to make Sarah’s and Matua’s piles equal (7+7=14). It usually takes time for the students to understand that Matua needs to give Sarah one sweet. The usual initial response is for the students to say that Matua should give two to Sarah.
Over the next 2-4 days the students work with a partner or in a small teaching group to solve problems involving near doubles. As the students solve the problems, encourage them to share their strategies with others.
At the end of each session ask volunteers to explain their working and thinking to the rest of the class. You might have students create charts demonstrating how they solved a problem using a doublind strategy. These could be added to throughout the week and shared at the end of each session, before being displayed at the conclusion of the unit.
Dear families and whānau
At school this week we are using our knowledge of doubles to solve number problems. Ask your child to explain their thinking as they solve these problems.
To work out 9 + 8, Marama says 8 + 8 = 16 and adds 1 to give 17.
Use Marama’s method to work out these:
To work out 8 + 12 Ariana takes 2 away from the 12 to leave 10 and adds this 2 to the 8 to give 10. So 8 + 12 is the same as 10 + 10 = 20.
Use Ariana’s method to work out these:
To work out 28 + 22, Lochie takes 2 away from the 22 to leave 20 and adds this 2 to the 28 to give 30. So 28 + 22 is the same as 30 + 20. So the answer is 50.
Using Lochie’s method to work out the following sums requires students to use mental strategies for addition and subtraction problems. See if you can use this strateggy to work out the answer to these problems:
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/smart-doubling at 8:31pm on the 26th February 2024